Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Counting how many ways 30 green balls... How many ways are there to distribute 30 green balls to 4 persons if Alice and Eve together get no more than 20 and Lucky gets at least 7?
The answer given to me was $2464 = C(26, 3) − 66 − 46-24$ but I got $C(26, 3) - 6$.
Here is what I did:
Using "stars and bars" (or whatever ... | Lucky gets $7$ balls right away. Alice and Eve can split their $j\in[0..20]$ balls in $j+1$ ways, then Lucky and Mike can split the remaining $23-j$ balls in $24-j$ ways. The total number of admissible allocations therefore is
$$\sum_{j=0}^{20}(j+1)(24-j)=2464\ .$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2722673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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An open set which does not belong to Uniform Topology of $R^w$ but belongs to $l^2$ topology of $R^w $ I can not find an open set which does not belong to Uniform Topology of $R^w$ but belongs to $l^2$ topology of $R^w $.
I know that $l^2$ topology of $R^w $ contains Uniform Topology of $R^w$. But I can not find those ... | As Nate Eldredge(https://math.stackexchange.com/users/822/nate-eldredge) mentioned , the unit ball i.e the set of all sequences $x$ satisfying $∑|x(n) |^2<1$ can be an example.
There are sequences of arbitrarily small uniform norm which are not contained in this ball.
So this is an open ball around the zero sequence... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The Modulus of all the roots of a Polynomial are equal to $1$
Suppose the real number $\lambda \in (0,1)$, and let $n$ be a positive integer. Prove that all roots of the polynomial $$f\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{k}$$ have modulus equal to $1.$
The Putnam problem 2014... | Write the above polynomial in the form: $a+bx+cx^2$... and observe that if we put $f(x)=0$ it is same as putting $x^k f(1/x)=0$ {property of binomial coefficients and the power of lambda} conclude that if $x$ is a root then $1/x$ is also has the same modulus. Hence $|x|=1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2722893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
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Image of a basis forms a basis, if and only if matrix is invertible Suppose $B_1=\{v_1,v_2,...,v_n\}$ is a basis of $\mathbb{R}^n$, and $M$ is an $n*n$ matrix. Prove that $B_2=\{Mv_1,Mv_2,...,Mv_n\}$ is also a basis of $\mathbb{R}^n$ if and only if $M$ is invertible.
Following is what I have so far:
Assume $B_2$ is bas... |
$\begin{bmatrix}Mv_1&Mv_2&...&Mv_n\end{bmatrix}=\begin{bmatrix}v_1&v_2&...&v_n\end{bmatrix}\begin{bmatrix}a_{11}&a_{12}&...&a_{1n}\\a_{21}&a_{22}&...&a_{2n}\\ \vdots&\vdots&\vdots&\vdots\\a_{n1}&a_{n2}&...&a_{nn}\end{bmatrix}$
The matrix on the right is just $M^T$.
Suppose you have a linear combination of the origin... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Matrix square roots of -I Since we can see matrices as generalizations of complex numbers, I asked myself if there is a way to classify those matrices which are the "Basis" for the complex part. That is, I would like to identify the set of $n\times n$ real valued matrices $M$ whose square $M^2$ is equal to $-I$, where ... | You might be interested in complex structures on real vector spaces.
The basic idea is this: suppose I have a complex vector space $V$ of dimension $n$. If I forget how to multiply by $i$, and only use scalars that are real, then $V$ is also a real vector space of dimension $2n$ (let's call this $V_{\mathbb{R}}$, to ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2723251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
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Period of the pendulum and taylor expansion The period of a (non-linear) simple pendulum is
$$
T(\theta_0) = \sqrt{8}/\omega_0 \int_0^{\theta_0} \frac{1}{\sqrt{\cos\theta-\cos\theta_0}}d\theta.
$$
Using elliptic functions, we can show that the term of order $1$ in $\theta_0$ is $2\pi/\omega_0$, which is precisely the p... | The issue is that $$\frac{\cos\theta-\cos\theta_0}{\theta-\theta_0}\approx -\sin\theta_0$$ only if $\theta$ and $\theta_0$ are close, otherwise that is not a good approximation, so you cannot recover the exact first term of the wanted Taylor series from it, since the value of the actual integrand function in a right ne... | {
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"source": "stackexchange",
"question_score": "1",
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On the asymptotic behavior of the Fourier coefficients of $(1-R\cos\theta)^{-3/2}$ Today in class I showed some ways for dealing with the classical integral $\int_{0}^{2\pi}\frac{d\theta}{(A+B\cos\theta)^2}$ under the constraints $A>B>0$, including
*
*Symmetry and the tangent half-angle substitution;
*Relating the ... | I don't think this is more elementary, but it may be a little more straightforward. Start with Heine's toroidal identity,
$$ \frac{1}{\sqrt{z-\cos\,t}} = \frac{\sqrt{2}}{\pi} \sum_{m=-\infty}^\infty\,Q_{m-1/2}(z)\,\exp{(i \, m\, t}), \, \, |z|\ge1, \, 0\le t \le 2\pi \, .$$
Splitting the summation into positive and n... | {
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"source": "stackexchange",
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Abelian group quotient computation New to group theory. Is the following correct? $a, b, c, d$ are independent elements generating the free abelian group $A = \langle a, b, c, d\rangle = \mathbb{Z}a \oplus \mathbb{Z}b \oplus \mathbb{Z}c \oplus \mathbb{Z}d$, and $B$ is the subgroup $B = \langle a, 2b-a, 2c-b, 2d-c\rangl... | In $A/B$, we have $c=2d$, $b=2c=4d$, $a=2b=8d$, $a=0$.
Therefore, $A/B$ is generated by the class of $d$ and this class has order $8$. Thus, $A/B \cong \mathbb Z/8 \mathbb Z$.
An explicit isomorphism is induced by the map $A \to \mathbb Z/8 \mathbb Z$ given by $\alpha a + \beta b + \gamma c + \delta d \mapsto 4 \beta +... | {
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A $5\times 5$ grid of single-digit numbers in $\mathbb N$, with one cell empty. What number should be in the cell?
16.$$\begin{array}{|c|c|c|c|c|}
\hline
2 & 7 & 4 & 3 & 5
\\\hline
7 & 3 & 4 & 5 & 4
\\\hline
1 & 3 & 2 & 2 & 6
\\\hline
2 & 4 & 5 & 4 & \mathbf{?}
\\\hline
8 & 3 & 6 & 3 & 5
\\\hline
\end{array} $$
Is the... |
$$\begin{array}{c}
x \\
y \\
p \\
q \\
r
\end{array} \qquad\to\qquad x^y = pqr\quad\text{(concatenated)} $$
"[W]hat does it really mean to solve such a question?"
Well, such a question challenges us to find order amid chaos. That's what mathematics ---as the study of pattern--- is all about, so it's not a completel... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Consider the ideal $I = \langle(3,4) \rangle$ of the ring $\mathbb{Z} ×\mathbb{Z}$. Prove that $(\mathbb{Z}×\mathbb{Z})/I$ is not a domain. Having some trouble with this. Need to show that there exist some Zero Divisor, but not really sure what that would look like in $(\mathbb{Z}×\mathbb{Z})/I$
Thanks in advance
| Hint: $(3,1) \cdot (1,4) = (3,4) \in I$.
| {
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Direct sum of two subspace Given the definition on textbook: Let $V$ be a subspace of $\mathbb R^n.$ Every vector $u \in \mathbb R^n$ can be written uniquely as $u = n + p.$ I still don't understand what it means because i am ask to do with a question that ask me the prove the uniqueness claim in the definition.
The q... | Let's look at an easy example.
Let $n=2$ and $V= <\begin{pmatrix} 1\\0 \end{pmatrix}>$. Then $V^{\perp}=<\begin{pmatrix} 0\\1 \end{pmatrix}>$. Now pick some $x\in\mathbb{R}^2$, say, $x=\begin{pmatrix} x_1\\ x_2 \end{pmatrix}$. Then $x=n+p$ where $n=x_1\begin{pmatrix} 1\\0 \end{pmatrix}\in V$ and $p=x_2\begin{pmatrix} 0... | {
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Prove using mathematical induction $1\cdot2+2\cdot2^2+3\cdot2^3+\ldots+n\cdot2^n=2[1+(n-1)2^n]$
Prove the result using Mathematical Induction $$1\cdot2+2\cdot2^2+3\cdot2^3+\ldots\ldots+n\cdot2^n=2[1+(n-1)2^n].$$
I've been stuck on this problem for hours, I have no idea how do even calculate it. The exponents throw me... | Hint:
The base case is fairly easy: $0\times\left(\ldots\right)=2\left(1+(0-1)\times2^0\right)$. If we assume that $\sum_{n=0}^k n2^{n}=2\left(1+(k-1)\cdot2^{k}\right)$, then our goal for the induction case is to prove that: $$
\underbrace{2\left(1+(k-1)\cdot2^{k}\right)}_{\sum_{n=0}^k {n2^{n}}}+ \underbrace{(k+1)2^{k+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2723987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $A$ is idempotent then $A$ is similar to a diagonal matrix with only $0$'s and $1$'s on the diagonal. I am trying to use Jordan normal form to show that if $A^2 = A$ then it is similar to a diagonal matrix with only $0$'s and $1$'s. I've proved that the eigenvalues of $A$ have to be either $0$ or $1$ so we know the ... | If $A^2=A$, then the equality should hold for each of the Jordan blocks. In more detail, $A=SJS^{-1}$. Then
$$
SJS^{-1}=A=A^2=SJ^2S^{-1},
$$
and then $J^2=J$. Now you can compare the individual Jordan blocks. In a Jordan block $J_1$, that is not diagonal and with eigenvalue $\alpha$, the $1,2$ entry is $1$. In $J_1^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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I need the steps to solving this limit without using l´Hopital rule I've tried many ways of solving this limit without using l'Hopital and I just can't figure it out. I know the answer is $3/2 \sin (2a).$
$$\lim_{x\,\to\,0} \frac{\sin(a+x)\sin(a+2x) - \sin^2(a)} x$$
Thank you!
| Perhaps not the most elegant, but:
$\lim_\limits{x\to0} \frac{\sin(a+x)\sin(a+2x) - \sin^2(a)} x\\
\lim_\limits{x\to 0} \frac{\frac 12 (-\cos(2a + 3x) + cos x) - \sin^2(a)} x\\
\lim_\limits{x\to 0} \frac{\frac 12 (-\cos(2a)\cos 3x + \sin(2a)\sin (3x) + \cos x) - \sin^2(a)} x\\
\lim_\limits{x\to 0} \frac{\frac 12 (-\cos... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simple Exclusion Process on a $100 \times 100$ checkerboard This is a problem given by a professor that has been perplexing me.
Suppose a particle takes a random walk on a $100 \times 100$ checkerboard in the following way. After an exponential time with rate 1, it attempts to move up, down, left, or right -- each wit... | A clever approach for the first part is to consider an infinite board ruled off in $100 \times 100$ sections. When the particle moves off the edge of your board, it moves to a space of the same type (corner or edge) of a neighboring board, but on the infinite grid all the cells are equivalent as there are no reflectio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2724403",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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Strange maths coincidence $(6\times 9)+(6+9)=69?$ Is this a freak coincidence in maths or there are more of this type of maths calculation?
$$\color{red}{(6\times 9)}+\color{blue}{(6+9)}=69$$
I try to find more, but I can't. Can you?
| $$(6 \times 9) + (6 + 9) = 10 \times 6 + 9 = 69$$
Similarly,
$$(n \times 9) + (n + 9) = 10 \times n + 9 = \text{"}n9\text{''},$$
for any one-digit number $n$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $\forall x,y \in B$ if $(x,y) \in S$ and if $x+y$ is even then $x=y$ then each class of $S$ has at most $2$ elements
Set $B = \{1,2,3,4,5\}$, $S$ - equivalence relation. It is given that for all $x,y \in B$ if $(x,y)\in S$ and if $x+y$ is an even number then $x = y$. In such case is it true that:
*
*the nu... | Assume that $S$ has an equivalence class with $2$ even numbers, namely $2$ and $4$. Then, $(x,y)\in S$ and $x+y=6$ which is even, but this contradicts $x=y$. Hence, two even numbers cannot be in the same equivalence class.
Now, assume that there exists an equivalence class with two odd numbers $x\neq y$. Then $x+y=\tex... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding eigenvalues and eigenvectors and then determining their geometric and algebraic multiplcities I have the following matrix:
$A = \begin{bmatrix}
1 && 7 && -2 \\
0 && 3 && -1 \\
0 && 0 && 2 \end{bmatrix}$
and I am trying to find the eigenvalues and eigenvectors followed by their respective geometric... | That RRE $\begin{bmatrix} 0&1&0\\ 0&0&1\\ 0&0&0 \end{bmatrix}$ means $y=0$ and $z=0$, so $$\begin{bmatrix} x\\y\\z \end{bmatrix} =\begin{bmatrix} x\\0\\0 \end{bmatrix} =x\begin{bmatrix} 1\\0\\0 \end{bmatrix}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2724790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Method of CDF for Y = 1/X I am trying to solve this question:
Let X be a standard cauchy variable. Define Y to be 1/X. I want to find the CDF of Y.
My problem: I am finding the CDF to be:
https://arachnoid.com/latex/?equ=%5Cfrac%7B1%7D%7B2%20%7D-%5Cfrac%7B1%7D%7B%5Cpi%20%7Darctan(%5Cfrac%7B1%7D%7By%20%7D)
But as a tak... | A standard trigonometric identity says
$$
\arctan \frac 1 y = \frac \pi 2 - \arctan y \text{ if } y>0
$$
so we get
$$
\frac 1 2 -\frac 1 \pi \arctan \frac 1 y = \frac 1 2 - \frac 1 \pi\left( \frac \pi 2 - \arctan y \right) \text{ if } y>0
$$
and this simplifies to
$$
\frac 1 \pi \arctan y
$$
Somewhere you should have h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2724974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Legendre polynomials and primality testing Can you provide a proof or a counterexample for the following claim ?
Let $n$ be an odd natural number greater than one . Let $r$ be the smallest odd prime number such that $r \nmid n$ and $n^2 \not\equiv 1 \pmod r$ . Let $P_n(x)$ be Legendre polynomial , then $n$ is a prime ... | See here for more information. This covers the test you are describing, however it is stated as a conjecture, not a primality test.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2725086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Neighbors have three children. Given three independent observations of boy, what is the probability that they have 3 boys? I would really appreciate some help on this one. I'm completely lost. I have no idea why my method doesnt work.
"Your new neighbors have three children. If you are told about three
independent o... | We can restate the question as follows. Suppose there are $3$ balls in an urn, and each ball is either black or white with equal probability. We draw one ball from the urn at random, observe the color, and replace it in the urn. Given that after $3$ draws we observed a black ball each time, what is the probability t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Matrix as tensor exercise with answer Can I get an explanation for the following exercise with answer in the book Linear Algebra via Exterior Products by S. Winitzki?
Exercise 1 - Matrices as Tensors. Now suppose you have a matrix $A_{jk}$ that specifies the linear operator $\hat A$ in a basis $\{\mathbf e_k\}.$ Which... | One of the basic insights of multilinear algebra is that "taking a vector" is in some sense equivalent to "giving you a covector" (because taking a vector is what that covector does), and conversely "taking a covector" is the same as "giving you a vector".
So we can view an $(1,1)$-tensor either as something that takes... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can we find the asymptotic behavior of this $f(x) =\int_{0}^{\infty}\frac{u^2}{1+\frac{e^{u^2}}{x}}du$? I encountered this function in Statistical Mechanics. $$f(x) =\int_{0}^{\infty}\frac{u^2}{1+\frac{e^{u^2}}{x}}du$$
For $x=0$, we define its value to be zero.
I wanted to see it's asymptotic behavior in the limit x ... | 1. (Not so illuminating) analytic expression. Assume for a moment that $0 < x < 1$. Then
\begin{align*}
f(x)
&= \int_{0}^{\infty} \frac{xu^2e^{-u^2}}{1 + xe^{-u^2}} \, du
= \sum_{n=1}^{\infty} (-1)^{n-1} x^n \int_{0}^{\infty} u^2 e^{-nu^2} \, du \\
&= -\frac{\sqrt{\pi}}{4} \sum_{n=1}^{\infty} \frac{(-x)^n}{n^{3/2}}
=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2725488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Consider the string $(∀x)(∀y)(f(x) = y → ((∀z)g(z) = f(x) ≡ (∀z)g(z) = y))$ Is this well-formed formula a tautology?
I am having trouble finding examples to help me work through these types of problems. I was hoping someone could help me out.
I think, using abstraction, this can be written as:
$$(∀x)(∀y)(p → (q ≡ q_1))... | The formula is a valid formula of First-order logic with equality.
Here is the derivation:
1) $f(x)=y$ --- premise
2) $(∀z)(g(z)=f(x))$ --- assumed [a]
3) $g(z)=f(x)$ --- from 2) by Universal instantiation
4) $g(z)=y$ --- from 1) and 3) bt transitivity of equality
5) $(∀z)(g(z)=y)$ --- from 4) by Generalization
6) $(∀... | {
"language": "en",
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Combining ratios of quantities with different sizes A friend and I had two different answers to this seeming simple question. It goes as follows:
Jar A contains flour and sugar in the ratio 5 : 1. Jar B, which is three times larger than Jar A, contains flour and sugar in the ratio 8 : 1. When the contents of these j... | I interpret "three times larger than" to mean "three times plus one the size of" instead of just "three times the size of". So, using a similar reasoning as Duchamp Gérard H. E.:
$$\text{Flour:} \quad (5/6)U+(8/9)4U=(5/6)U+(32/9)U=(79/18)U\\ \text{Sugar:} \quad (1/6)U+(1/9)4U=(1/6)U+(4/9)U=(11/18)U.$$
The ratio $(79/18... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2725658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Solve $\tan (\theta) + \tan (2\theta) = \tan (3\theta)$
Find the general solution of:
$$\tan (\theta) + \tan (2\theta) = \tan (3\theta)$$
My Attempt:
$$\tan (\theta) + \tan (2\theta) = \tan (3\theta)$$
$$\dfrac {\sin (\theta)}{\cos (\theta)}+ \dfrac {\sin (2\theta)}{\cos (2\theta)}=\dfrac {\sin (3\theta)}{\cos (3\t... | I'll give a hint to get you started which is that $\text{tan}(a + b) = \dfrac{\text{tan}(a) + \text{tan}(b)}{1-\text{tan}(a)\text{tan}(b)}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2725922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove that $\bigcup S_n = (0,1.5)$ if $S_n=\{\frac{1}{n}\le x< 1+\frac{1}{n}\}$?
Given that $S_n=\{x\in \mathbb R|\frac{1}{n}\le x< 1+\frac{1}{n}\}$, $n\in \mathbb N - \{0,1\}$ I need to find $\bigcup S_n$.
If we plug in $n=2$ then $0.5 \le x < 1.5$.
If $n\to \infty$ then $0<x\le 1$. Thefore $\bigcup S_n=(0,1.... | For what it is worth, I was wondering if there was a way to calculate the 1.5 instead of guessing it first, and I found the following proof. This may or may not be to your taste; see e.g. EWD1300 if you are interested in the background of this style of proof and notation.
In this answer $\;x\;$ is real and $\;n\;$ is ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Finding complex roots of fourth degree polynomial $z^4 + 8z^3 + 16z^2 + 9$ I have the equation: $$z^4 + 8z^3 + 16z^2 + 9 = 0$$
I need to find all the complex solutions and I've got no clue how to approach it. I've tried factoring but nothing came out of it. I'm still very new to the world of complex numbers so I'll ap... | If you substitute $$x=z+2$$ the equation turns into $$x^4-8x^2+25=0$$ which can be solved by a further substitution $y=x^2$
| {
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"url": "https://math.stackexchange.com/questions/2726195",
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"source": "stackexchange",
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$\mathcal{O}_K$ UFD $\iff h_K=1$ How can we prove that, if $K$ is a number field, then his integer ring $\mathcal{O}_K$ is an unique factorization domain if and only if the class number of $K$ is 1?
| Consider these:
*
*The ideals of $\mathcal{O}_K$ have unique factorization into prime ideals.
*The class number of $K$ is $1$ iff every ideal of $\mathcal{O}_K$ is principal.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find a 90% confidence interval for the population variance The variability in the amount of impurities present in a batch of chemicals used for a particular process depends on the length of time that the process is in operation. The standard deviation for the traditional process is 1.24. A new process has been develope... | Seems to me you didn't quite finish. I believe you want a 2-sided confidence interval.
If $L$ and $U$ cut 5% from lower and upper tails of $\mathsf{Chisq}(n-1),$ respectively, then from your first displayed equation a 90% CI for $\sigma^2$ is of the form $\left((n-1)S^2/U,\, (n-1)S^2/L\right).$
| {
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Find value of $t$ where slope of parametrically-defined curve $=4$ using multivariable calculus A recent problem I encountered gave me curve $C$ defined by the parametric equations $x(t)=2t^{2}+t-1$ and $y(t)=t^{2}-3t+1$, and asked me to find point $t$ where the slope of the tangent line to $C$ would equal $4$. Obvious... | Hint: you will get the equation: $$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{2t-3}{4t+1}=4$$. Solve this for $t$!
| {
"language": "en",
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Show that this ring has no identity.
Let $R=\left \{ g:\Bbb{R}\to \Bbb{R} \mid g \text{ is continuous and } g(1)=0 \right \}$ be a ring. Show that $R$ has no identity.
The answer says there does not exist a function $h(x)\in R$ such that $h(x)=1$, which I don't understand why, since the only condition in $R$ is $g(1)... | *
*The most obvious choice of identity function is the constant function $h(x) = 1$. After all, multiplying $h(x)$ pointwise by another function $g(x)$ results in $g(x)$ again.
*Unfortunately, $h(x)$ doesn't belong to the ring $R$ because $h(1)=1\neq 0$. It can't be the identity because it isn't in $R$.
*We can try... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to show that the set $Z(G)=\{z \in G : \forall g \in G, z * g = g * z\}$ is a subgroup of $(G, *)$? I have shown that the neutral element is in $Z(G)$. I have also shown that the law is closed in $Z(G)$.
However, I'm not sure how to prove that $\forall x \in Z(G), \exists x' \in Z(G)$ such that $x * x' = x' * x = ... | If $x\in Z(G)$, there is a $x'\in G$ such that $xx'=x'x=e$. So, the problem is to prove that $x'\in Z(G)$. Take $g\in G$. Now, take $g'\in G$ such that $gg'=g'g=e$. Then $xg'=g'x$ (since $x\in Z(G)$), but it follows from this that $gx'=x'g$, as we wanted to prove.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Perigee, Apogee Upper Bounds from Belaga and Mignotte I am currently reading the work of Belaga on upper bounds on minimal cyclic iterates in the $3x+d$ problem.
In the paper, the author gives an upper bound on the perigee as
$$
dk^{C_2}
$$
where $k$ is the number of odd elements in the cycle, and $C_2$ is an ef... | One can apply the results of Rhin (as provided by Lemma 12 in the work of Simons and De Weger) to derive sharp constants.
Assume $k+l>k$. Lemma 12 in Simons/De Weger demonstrates the inequality
$$ (k+l)\log 2 - k\log 3 > e^{-13.3(0.46507)}k^{-13.3}.$$ This inequality provides means for deriving a lower bound on the ... | {
"language": "en",
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If $p>0$, then $ \lim_{n\to\infty}\frac{1}{n^p}=0$ using squeeze theorem for sequences.
If $p>0$, then $ \lim_{n\to\infty}\frac{1}{n^p}=0\;.$
Rudin suggests in his Principle of Mathematical Analysis to take $$n> (\frac{1}{\epsilon})^\frac{1}{p}$$ using the Archimedean property of the real number system.
This is unde... | What you mentioned is Theorem 3.20 in Rudin's book (page 57).
If one takes $n>(1/\varepsilon)^{1/p}$, then it follows that $n^p>1/\varepsilon$ (because $p>0$) and thus $\displaystyle\frac{1}{n^p}<\varepsilon$. In particular, this argument shows that if one takes a positive integer $N>(1/\varepsilon)^{1/p}$, existence g... | {
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Is the set of all Irrational Numbers a ring or a field? I would really appreciate a proof of either one.
I think it should be a field as it satisfies the multiplicative and additive identities and is commutative.
| A ring requires a valid operation (addition) with an inverse (subtraction) and an identity or zero satisfying $x+0=x$ for all $x$.
Likewise a field requires a further operation (multiplication) which is distributive over addition (i.e. $a(b+c)=ab+ac$) and which also has an identity or unit satisfying $1\times x=x$ for ... | {
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Number of roots of the equation $x\sin x-1=0$ for $x\in [0,2\pi]$ Number of roots of the equation $x \sin x-1=0$ for $x\in [0,2\pi]$
My attempt is using the bisection method
where I initially took $a=0$, $b=\frac{\pi}{2}$
as $f(a)\cdot f(c)<0$
We can proceed along the lines
we get one root $\in[0,\frac{\pi}{2}]$
Sim... | Here is the plot of f(x) = $x \sin(x) - 1$ for $0\le x \le 2\pi $.
If $f(a)f(c)\lt0$ there must be at least one root between $a$ and $c$ but there could be more!
Also if $f(a)f(c)\gt0$, it does not mean that there is no root between $a$ and $c$. In fact, you could have infinite number of roots there. There is no simp... | {
"language": "en",
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How to find the value of $x$ that satisfies $3x=4$ in $\mathbb Z/5\mathbb Z$? Let $\mathbb Z_5 = \mathbb Z/5\mathbb Z$.
The value of $x$ which satisfies the equation $3x = 4\bmod 5$ is...?
The answer is $3$. I understand why the answer is $3$, but not how it was derived. Is there an equation or process I can use that w... | Consider equations of the form $m\cdot x = n\bmod p$, where $p$ is a prime number.
The other answer is a correct way to derive the solution, but it takes as many as $p$ tries to find it in the worst case. A more efficient solution is the following. First, compute $m^{-1}$, that is, the only solution $y$ to the equation... | {
"language": "en",
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Parametric equation of a curve: a line in a circle transform to a curve in an ellipse Giving the circle with a line segment inside, if the circle was stretched into an ellipse, what is the parametric equation of the parabolic curve (I assume) transformed from the line segment?
circle to ellipse
I want the curve as in s... | How do you define your stretch transformation? In any transformation a relation between variables before and after transformation is defined.
Setting a variable in one system to a constant we can map the curve in the other system. Like eg polar to cartesian coordinates $ r=a \rightarrow x^2+y^2= a^2. $
I could not fi... | {
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Function’s graph - a homework problem Does there exist an $f(x)$ function $R\to R$, such that every line parallel to the $x$-axis meets the function’s graph
a) exactly three times?
b) an even number of times?
For b), I have several examples, which almost work, but there is a line which is tangent to the curve, so it do... | Hint: For (a), consider $f(x)$ on a single interval of $\mathbb{R}$. You could try partitioning $f(x)$ into $3$ sections over this interval. You could then repeat this in every other interval but using a different range of $y$-values. You could use a different number of partitions for (b). An example function is shown ... | {
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Why this is wrong let, n be a positive integer.Then,${C_r}=$$ {n} \choose {r}$.Now evaluate $ {C_0}- {C_1}/2+ {C_2}/3+.....+ (-1)^n {C_n}/(n+1)$
I expand $(1-x)^n$ and integrating both side and putting $x=1$ the required series comes.But it gets 0.But answer is $1/n+1$
| Another way:
$$\dfrac{\binom nr}{r+1}=\cdots\dfrac{\binom{n+1}{r+1}}{n+1}$$
$$\sum_{r=0}^n\dfrac{(-1)^r\binom nr}{r+1}=\dfrac1{n+1}\sum_{r=0}^n\binom{n+1}{r+1}$$
Now $$\sum_{r=0}^n(-1)^r\binom{n+1}{r+1}=\binom{n+1}0-(1-1)^{n+1}$$
| {
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Number of $6$-digit numbers made up of the digits $1$, $2$, and $3$ with no digit occurring $3$ or more times consecutively? Find the number of 6-digit numbers made up of the digits $1$, $2$, and $3$ that have no digit occur three or more times consecutively. (For example, $123123$ would count, but $123111$ would not.)... | There are $3^6 = 729$ possible sequences. From these, we must subtract those in which three consecutive digits are the same.
Observe that a prohibited sequence must begin in one of the first four positions. Let $A_1, A_2, A_3, A_4$ be the set of outcomes in which three consecutive digits beginning in the first, secon... | {
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Evaluating $\prod^{100}_{k=1}\left[1+2\cos \frac{2\pi \cdot 3^k}{3^{100}+1}\right]$
Evaluate$$\prod^{100}_{k=1}\left[1+2\cos \frac{2\pi \cdot 3^k}{3^{100}+1}\right]$$
My attempt:
$$1+2\cos 2\theta= 1+2(1-2\sin^2\theta)=3-4\sin^2\theta$$
$$=\frac{3\sin \theta-4\sin^3\theta}{\sin \theta}=\frac{\sin 3\theta}{\sin \thet... | Let $$z=\cos\bigg(\frac{2\pi}{3^n+1}\bigg)+i\sin\bigg(\frac{2\pi}{3^n+1}\bigg)$$
Then $z^{3^n+1}=1$ and also $\displaystyle 2\cos \bigg(\frac{2\pi\cdot 3^k}{3^n+1}\bigg)=z^{3^k}+\frac{1}{z^{3^k}}$
Write $$\prod^{n}_{k=1}\bigg[1+2\cos\bigg(\frac{2\pi\cdot 3^k}{3^n+1}\bigg)\bigg]$$
$$=\bigg(1+z^3+\frac{1}{z^3}\bigg)\bigg... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Integral of $ \int \frac {\tanh(x) dx}{\tanh(x)+\operatorname{sech}(x) }$ My attempt at solution:
Also I got a question, is there a way this can be solved without using hyperbolic tangential half angle substitution? Because I don't get how you can treat hyperbolic functions as if they were trigonometric and deducing s... | The integral can also be seen as the following:
\begin{align}
\int \frac{\tanh(x) \, dx}{\tanh(x) + sech(x)} &= \int \frac{\sinh(x) \, dx}{\sinh(x) + 1} = \int \left( 1 - \frac{1}{1 + \sinh(x)} \right) \, dx \\
&= x + \sqrt{2} \, \tanh^{-1}\left( \frac{1}{\sqrt{2}} \, \left(1 - \tanh\left(\frac{x}{2}\right) \right) \ri... | {
"language": "en",
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Why is the value of the series representation for $\arctan$ at $ x = 1$ necessarily $\arctan(1)$ Exercise 6.6.1. The derivation in Example 6.6.1 shows the Taylor series for $\arctan(x)$ is valid for all $x \in (−1,1)$. Notice, however, that the series also converges when $x = 1$. Assuming that $\arctan(x)$ is continuou... | Proof. Since $arctan(x)$ is continuous, for all $\epsilon > 0 $ there exists some $\delta > 0 $ such that $|1-x| < \delta $ implies that $|arctan(1) - arctan(x)| < \epsilon $. We also know that $$arctan(x) = x - \frac 1 3 x^3 + \frac 1 5 x^5 - \frac 1 7 x^7 + \cdots $$ when $ x \in (-1, 1)$. Hence $|1-x| < \delta $ im... | {
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Why is the integral defined as the limit of the sum $\int_a^b f(x) dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\Delta x$? I am failing to understand why the integral is defined as:
$$\int_a^b f(x) dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\Delta x$$
instead of:
$$\int_a^b f(x)dx=\sum_{i=1}^\infty f(x_i^*)\Delta x$$
Is... | Actually, the definition is:
Let $\mathcal{P}=\{a, x_1, \ldots, x_k, b\}$ be a partition of $[a, b]$ and denote $\Delta \mathcal{P} = \max_i |x_i-x_{i+1}|$. Then we say a bounded function $f$ is Riemann integrable provided for any $\varepsilon>0$ there exists $\delta>0$ such that if $\Delta \mathcal{P}<\delta$ implie... | {
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Union of two compact totally disconnected sets in $\mathbb{R}$ is totally disconnected I'm so stuck with the next problem. First, the definition that I have are the next:
$X$ is totally disconnected if for all $x\in X$ we have that $C_x=\{x\}$ where $C_x$ is the connected component.
Let $A,B\subseteq\mathbb{R}$ be a c... | First of all recall that a subset of $\mathbb{R}$ is connected if and only if it is an interval. The same holds for intervals as well: a subset of an interval is connected if and only if it is again an interval. Here by interval I understand a subset $I\subseteq\mathbb{R}$ such that if $a,b\in I$ and $a<c<b$ for some $... | {
"language": "en",
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The image of $\emptyset$ under $f$ is empty. Let A= $\{ \emptyset,\{1\},\{2,\},...,\{n\} \}$. Define $f\colon A \to \mathbb{N}$ such that $f(\emptyset)=0$ and $f(\{n\})=n$.
Is this a counterexample that the image of the $\emptyset$ under an arbitrary function $f$ is empty?
| You are defining a function on a set $X$ that has $\emptyset$ as an element. So $f(\emptyset)$ can be anything you like.
But you're probably thinking of the notion of "image of a subspace under $f$".
I (and many texts/papers) always separate these notions notationwise, using square brackets so $f[\emptyset] = \emptys... | {
"language": "en",
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Angle trisection of $90^o$ Read on page #7 of article here, that angle of $90^o$ can be trisected. I went through this youtube video here, and here; and denote these two videos (denoting two separate methods) by (a), (b) respectively. These use elementary methods used in school days, but were never explained for the re... |
The line $AB$ is perpendicular to $BC$. The circles have, respectively, centers $B$ and $C$ and radius $BC$. The segments $BD$, $BC$ and $CD$ are equal by construction. Can you prove that the angle $\widehat{CBD}$ is twice the angle $\widehat{DBA}$?
| {
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$f$ convex, $g$ concave and increasing, $\int_0^1 f = \int_0^1 g$, then $\int_0^1(f)^2 \geq \int_0^1(g)^2$
Let $f,g:[0,1] \to [0, \infty)$ be two continuous functions such that $$f(0) = g(0) = 0,$$ $f$ is convex, $g$ is concave and increasing and $$\displaystyle \int_0^1f(x)dx = \int_0^1g(x)dx.$$ Prove that $$ \displa... | For some $0 < \epsilon < 1$,
$$
\int_\epsilon^1 (f(x)^2-g(x)^2)dx = \int_\epsilon^1 (f(x) + g(x))(f(x)-g(x)) dx
$$
since $f(x)+g(x) \ge 0$, we can apply the intermediate value theorem to get a $\xi \in [\epsilon,1]$, such that
$$
\int_\epsilon^1 (f(x) + g(x))(f(x)-g(x))dx = (f(\xi)+g(\xi))\int_\epsilon^1 (f(x)-g(x))dx ... | {
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"timestamp": "2023-03-29T00:00:00",
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Proof that series converges in probability. Consider $X_{1} \dots X_{n} \dots$ - independent random variables with Cauchy-distribution with scale parameter $\theta_{n}$ and zero location parameter such as : $\sum_{n \ge 1} \theta_{n}$ converges. Prove that $X = \sum_{n \ge 1} X_{n}$ converges in probability.
My attempt... | By Cauchy criterion for convergence in probability it suffices to prove that $$\lim_{n,m\to \infty}P\left(\left|\sum_{k=n+1}^m X_k \right| > \epsilon \right) = 0$$
A simple computation with characteristic functions proves that $\sum_{k=n+1}^m X_k$ follows a Cauchy distribution with location $0$ and scale parameter $\su... | {
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Compute the limit of the sequence ${\textstyle\sum_{k=1}^n}\frac1{\sqrt{n^2+k}}$ I have to compute the limit of this sequence ${\textstyle\sum_{k=1}^n}\frac1{\sqrt{n^2+k}}$ as $n\rightarrow\infty$. First I was thinking about some Riemann sum and and forced the $n^{2}$ outside the square root but the function was not so... | How about squeezing ?
$$\frac{n}{\sqrt{n^2+n}}\leq \sum_{k=1}^n\frac1{\sqrt{n^2+k}}\leq \frac{n}{\sqrt{n^2+1}}$$
The outer terms both go to $1$.
| {
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Graph Theory - proving equivalence
Let $G$ be a graph with order $n ≥ 2$. Prove that $G$ satisfies $(a)$ if and only if it satisfies $(b)$.
$(a)$ for any pair of distinct vertices $u, v$ of $G$, there exists at least one $(u, v)$-path in $G$;
$(b)$ for any partition ${X, Y }$ of $V (G)$ into two non-empty parts $X, Y$... | You made use of the word "connected" all too quickly. We have to get hold of this notion first. Call two vertices $u$,$v\in V$ equivalent if there is an edge path in $G$ connecting $u$ and $v$.
Make sure that this is indeed an equivalence relation on the set $V$ of vertices of $G$. How many equivalence classes can ther... | {
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"url": "https://math.stackexchange.com/questions/2729616",
"timestamp": "2023-03-29T00:00:00",
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Let $S$ be a metric space, $ f: S \to \Bbb R$ continuous. Define $Z(f) = \{p \in S : f(p) = 0 \}$. Prove $Z(f)$ is closed. Let $S$ be a metric space, $ f: S \to \Bbb R$ continuous. Define $Z(f) = \{p \in S : f(p) = 0 \}$. Prove $Z(f)$ is closed.
I've come up with a proof... I just would like to know if it is logical en... | Another way is to use sequential characterization, so let $(p_{n})\subseteq Z(f)$ be such that $p_{n}\rightarrow p$ in $S$, then continuity of $f$ gives $f(p_{n})\rightarrow f(p)$. But $f(p_{n})=0$, so $f(p)=0$, this shows that $p\in Z(f)$, we are done.
| {
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} |
Showing the sequence converges $a_{1}=\frac{1}{2}$, $a_{n+1}=\frac{1}{2+a_{n}}$ Showing the sequence converges $a_{1}=\frac{1}{2}$, $a_{n+1}=\frac{1}{2+a_{n}}$.
I already know that if $(a_{n})$ converges then it does to $\sqrt{2}-1$.But i dont't know how to prove that this sequence cenverges.
EDIT
I think that the su... | Since you already have a reasonable conjecture consider the sequence $(b_n)_{n\geq1}$ defined by
$$a_n=\sqrt{2}-1+b_n,\qquad{\rm resp.,}\qquad b_n=a_n+1-\sqrt{2}\qquad(n\geq1)\ ,$$
and try to prove that $\lim_{n\to\infty} b_n=0$, which should be simpler.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Problem Understanding a Definition of Order of a Field I need some help regarding some definitons.
I was studying algebraic number theory and I am stuck on this.Can someone explain me what is meant by $F = [\mathcal{O}_k:\mathcal{O}]$ with an easy examples
I was studying the following theorem but didn't get examples
Le... | Since you mentioned that you're studying algebraic number theory, I'm assuming $K$ is an algebraic number field in your context.
An order $\mathcal{O}$ of a number field $K$ is defined to be a subring of the ring of integers $\mathcal{O}_K$ with $d$ generators over $\mathbb{Z}$, where $d$ is the degree of the number fi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Dividing polynomials in $\mathbb{F}_7[x]$ I am trying to divide $4x^4+3x^3+2x^2+x+1$ by $2x^2+x+1$ in $\mathbb{F}_7[x]$.
Normally outside of $\mathbb{F}_7[x]$ I know that the answer would be $2x^2+(1/2)x-(1/4)$ with a remainder of $(3/4)x+(5/4)$. But because this is in $\mathbb{F}_7[x]$ and the coefficients must be in ... | $\frac{1}{2}=\frac{1×4}{2×4}=\frac{4}{8}=\frac{4}{1}$ as 8 mod 7 is 1.Here you need to make denominator 1 by selecting suitable number.
$\frac{-1}{4}=\frac{6×2}{8}=\frac{6×2}{1}=\frac{5}{1}$. Since $-1=6 mod 7$.
Similarly you can try other fraction mod 7.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2730083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Show that if the integral of the "derivative" of a L^1 function is 0, then the function is constant a.e. I'm working on the following problem:
Let $f\in L^1[a,b]$. Prove that if
$$\lim\limits_{h\to 0}\frac 1h\int_a^b|f(x+h)-f(x)|dx=0,(*)$$
then there is a constant $c$ such that $f(x)=c$ for almost every $x\in (a,b)$.... | This is an easy consequence of Lebesgue's Theorem. Let $a<c<d<b$ and assume that $c$ and $d$ are Lebesgue points of $f$. Then $\frac {\int_c^{d} f(x+h)dx -\int_c^{d}f(x)dx} h \to 0$. This gives $\frac {\int_d^{d+h} f(x+h)dx -\int_c^{c+h}f(x)dx} h \to 0$. Since $c$ and $d$ are Lebesgue points this gives $f(c)=f(d)$. Sin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove the $\ell^2$ norm of a linear transformation $A: \mathbb{R}^n \to \mathbb R^n$ is the maximum eigenvalue If $A: \mathbb R^n \to \mathbb R^n$ is a linear transformation and $\mathbb R^n$ is equipped with $\lVert \cdot \rVert_2$, prove that
$$ \lVert A \rVert := \sup \left\{ \frac{\lVert A \vec{x} \rVert_2}{\lVert ... | The transformation $T(1,0)=(1,0), T(0,1)=(1,1)$ is non-singular has only 1 as its eigen value but $||T||=\sqrt 2$. The stated equality holds for symmetric matrices (even singular ones!) but the result claimed is false.
| {
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"url": "https://math.stackexchange.com/questions/2730366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability Random selection from class
*Suppose that a certain college class contains $62$ students. Of these, $35$ are sophomores, $38$ are biology majors, and $12$ are neither. A student is selected at random from the class.
(a) What is the probability that the student is both a sophomore and a biology major?
(b... | I agree with @MattiP that your answers are correct. But I want to mention
a general method for problems like this. (Making a table of the kind I
suggest is also useful for somewhat more advanced topics in statistics.)
It is often useful to make a $2 \times 2$ table for such situations:
You are given this information:
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Decaying rate of a convolution between an integrable function and a Schwartz function Suppose $f\in L^1(R^n)$ and $g\in S(R^n)$, where $S(R^n)$ is Schwartz space. Then, Can I have estimation like following?
$$
|[f*g](x)|\leq\frac{1}{(1+|x|)^{s}},
$$
for some $s>n$.
If it is correct, how to prove it? If it is not correc... | Of course the answer to the question as stated is "of course not"; the sensible version of the question is whether we have $$
|[f*g](x)|\leq\frac{c}{(1+|x|)^{s}}.
$$
The answer to that question is still no, although it's not so obvious.
Take $n=1$ just to simplify the notation. Choose $g\in\mathcal S(\Bbb R)$ with $g\g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2730651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The inexplicable approach of an Indian mathematician for Cosecant I am reading about the old Indian mathematician approximation
$$\csc (z)\simeq \frac{z^4+\pi ^2 z^2+2 \pi ^4}{2 \pi ^4 z-2 \pi ^2 z^3}$$
reminiscent of Bhaskara's.
I tried to use Taylor series and I got something similar to a Padé approximation
$$\csc (... | The Indian approximation encodes a large portion of the Weierstrass product for the sine function:
$$ \sin(z) = z\left(1-\frac{z^2}{\pi^2}\right)\prod_{n\geq 2}\left(1-\frac{z^2}{n^2\pi^2}\right) \tag{1}$$
The problem of approximating $\csc(z)$ boils down to producing approximations for
$$ g(z)=\frac{z}{\sin z}\left(1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2730733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove $\|A\| \leq \|A\|_{HS}$, where $\|A\|$ is the operator norm of A I Am trying to solve the following problem Let
$H_1
,H_
2$
be Hilbert spaces. Let
$A
\in B(H_1
,H_2
)$ be a Hilbert-Schmidt operator. For a complete
orthonormal sequence $(
u_n
)$ in
$H_1$, define the
Hilbert-Schmidt norm
$\|.\|_{HS}$
by... | In the last inequalities you have used an inequality of the form:
$$\left\Vert\sum_n a _n b_n \right\Vert \leq \left\Vert\sum_n a _n \right\Vert \left|\sum_n b_n \right|$$
which is not true. However you can use a Cauchy-Schwartz inequality which gives:
$$\sum_n \left\Vert a _n\right\Vert |b_n| \leq \left(\sum_n \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2730851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is the contrapositive of this simple proposition?
If $a$ and $b$ are rational then $ab$ and $a+b$ are also rational.
What is the contrapositive of this proposition?
*
*If $ab$ and $a+b$ are not rational then $a$ and $b$ are not rational.
*If $ab$ or $a+b$ are not rational then $a$ or $b$ are not rational.
| Your statement: $a$ and $b$ rational $\implies$ ($ab$ and $a+b$) rational.
Contrapositive: not ($a+b$ and $ab$ rational) $\equiv$ not ($a+b$ rational) or not ($ab$ rational) $\implies$ not($a$ and $b$ rational) $\equiv$ not ($a$ rational) or not ($b$ rational).
(My initial answer had a silly mistake in it)
| {
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Show that if $Y$ is path connected, then there is only one homotopy class of maps of $[0,1]$ into Y This question is taken from Mukres' Topology (exercise 51.2.b). This site provides the following answer:
All “closed” segments of a path are homotopic to the whole path. Since $Y$ is path connected, for any two paths we... | Write $I=[0,1]$. Then there is a homotopy between the constant
map $I\to I$ and the map $I\to I$ taking the whole of $I$ to $0$. Composing this with any
continuous map $f:I\to Y$ gives a homotopy between $f$ and the constant
map taking $I$ to $f(0)$. So every map from $I$ to $Y$ is homotopic
to a constant map. If $Y$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2731100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove the following sequence diverges: $\sqrt{n}-\frac{1}{n^2}+4$ I am trying to work through proving some sequences diverge. I am having a really hard time with the inequality arguments and I'm not sure why. The current problem is proving that $$\sqrt{n}-\frac{1}{n^2}+4$$ diverges to infinity.
I understand that essent... | I think that I got it thanks to dem0nakos comment about not needing the best possible $N$.
Proof:
Let $c$ be any positive number. By the Archimedean property we can select a natural number $N_1$ so that $N_1>4c^2$ and therefore $\sqrt{N_1}>2c$. Simultaneously we can find an $N_2$ such that $\frac{1}{N_2^2}<c$. If we le... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Tricky Graph Theory Puzzle I ran into an intriguing puzzle on Reddit that I thought could use some attention.
You start with a 3x3 grid labeled with numbers 1-9 like this:
$$ 1 \text{ }\text{ }\text{ }2\text{ }\text{ }\text{ }4
\\ 5 \text{ }\text{ }\text{ }6\text{ }\text{ }\text{ }8
\\ 9 \text{ }\text{ }\text{ }3\text{... | Condition 4: Not being allowed to go back over edges definitely changes the complexion of the problem. In fact, for ALMOST ALL numberings of an $n \times n$ grid as $n$ gets large, such a completion of the puzzle is impossible.
Why? The resulting digraph on the $n \times n = n^2$ vertices (vertices here taken as point... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2731356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How can I define closeness of these geometric shapes Given points on a 2D plane, what kind of metrics can be used to define if they closely fit either:
*
*triangle
*square or rectangle
*circle
*oval (circular but not oval)
(Image credit: StyleCraze.com "How to Determine the Shape of Your Face".)
Note thanks... | The four shapes given are all examples of (boundaries of) 2-dimensional convex bodies, so any metric on arbitrary convex bodies will do. Some examples include the following, where $C,D$ are $d$-dimensional convex sets in Euclidean space:
*
*Hausdorff metric: $d(C,D)= \max\left\{\sup\limits_{x \in X}\inf\limits_{y \i... | {
"language": "en",
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Show that $p(x) = a_nx^n+a...+a_1x+a_0$ either has a root or attains minimum value in R. Show that if $p(x) = a_nx^n +\dots+ a_1x + a_0$ and $a_n > 0$, then either $p(x) = 0$ has a solution, or else $p(x)$ has attains minimum value on $\mathbb{R}$.
I'm sorry I don't know how to even start the problem. I know that if it... | Hint: Recall that polynomials are continuous on $\Bbb R$; note that $p(x)\to\infty$ as $x\to\infty$; if $p(x)\lt 0$ for some $x=a$, apply IVT to $p$ on $[a,\infty)$; if $p(x)\gt 0$ for all $x$, recall that $\Bbb R$ has the glb property.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Question on irreducibility of $x^{nm}-a$ when $n$ and $m$ are coprime Let $F$ be a field and $a\in F$ and let $m,n$ be coprime positive integers such that. Then $x^{mn}−a$ is irreducible in $F[x]$ if and only if both $x^m−a $ and $x^n−a$ are irreducible in $F[x]$. This is supposed to be a Galois Theory question howeve... | This is an exercise on irreducibility, not on splitting fields. If $X^{mn}-a$ is irreducible over $F$, so are $X^{m}-a$ and $X^{n}-a$ because $X^{mn}-a=(X^{m})^n-a=(X^{n})^m-a$.
Conversely, assume $X^{m}-a$ and $X^{n}-a$ irreducible over $F$, and let $\alpha$ be a root of $X^{mn}-a$ in an algebraic closure. Then $\alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2731695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Showing the kernel of a set $X\subset \Bbb R^2$ is convex
Let $X\subset \Bbb R^2$. We define $V(x)$ to be the set of points in $X$ that $x$ 'can see', i.e. $V(x)=\{y\in X\mid [x,y]\subset X\}$.
The kernel is then $\text{ker}(X)=\{x\in X\mid V(x)=X\}$.
I want to show that the kernel is a convex set.
This is an exercis... | If $a,\ b\in {\rm Ker}\ X$ and $z\in X$, then $$ [az],\ [bz]\subset
X $$
If $c=ta+(1-t)b,\ 0<t<1$, then assume that $[cz]$ is not in $X$.
That is, there is $x\in (cz)$ not in $X$.
Since $[az]\subset X$, so $[sa+(1-s)z\ b]\subset X$ by assumption on
$b$.
For suitable $s$, $[sa+(1-s)z\ b]$ contains $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2731822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What's that process called where you form new group law by $x \star y = x \cdot a \cdot y$ for some $a$ in the group? Let $A$ be an abelian group. We can form new groups $(A, \cdot a \cdot)$ where $a$ is any element of $A$. Choosing $a = 1$ the identity of $A$ gives $A$ itself.
Clearly, associativity comes from assoc... | I have seen this also for groups, but it seems to be more interesting for Lie algebras, see here and here. For groups it seems to be called variant, see the comments at the above questions, or a sandwich, e.g., Semigroups under a sandwich operation, Proc. Edinburgh Math. Soc. (Ser. 2) 26 (1983), 371-382.
For $K$-algebr... | {
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For compass and straightedge problems, are you allowed to use the compass as a ruler? For compass and straightedge problems, you could have a line between two points A and B, and want to make a line the same size between C and line DE.
If you placed the two points of the compass between A and B, and made a circle aroun... | Yes. Not by the rules about how to use compass and straightedge but because it can be proved that it's as if we could do it (that's proposition 2 of book I of Euclid's Elements).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2732103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$C^*$-subalgebra of $M_2(\mathbb{C})$ Consider the following subalgebra of $M_2(\mathbb{C})$:
$$A=
\left\{\begin{bmatrix}
a & b \\
b & a
\end{bmatrix} : a,b\in \mathbb{C}\right\}.$$
One method I know that $A$ is isometrically *-isomorphic to $C(K)$, where $K=\{1,2\}$. (Because we can assume function $f$ such th... | Suppose that $\pi:A\to\mathbb C$ is a homorphism.
If $\pi(I)=0$, then $\pi=0$. Otherwise, from $\pi(I)=\pi(I^2)=\pi(I)^2$, we deduce that $\pi(I)=1$. In any case, $\pi(aI)=a$. Also, since
$$
x=\begin{bmatrix} 0&1\\1&0\end{bmatrix}
$$
satisfies $x^2=I$, we deduce that $\pi(x)^2=1$. So $\pi(x)=1$ or $\pi(x)=-1$. And t... | {
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"timestamp": "2023-03-29T00:00:00",
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Problem on normed space. Let $X$ be a normed space, $Y$ a dense subspace of $X$ and $Z$ a closed finite-codimensional subspace of $X$. Is $Z\cap Y $dense in $Z$ ?
I have no idea how to solve this problem. I am using this website for the first time, any help would be appreciated.
| Suppose first that the codimension of $Z$ is $1$. Then $Z=\ker\phi$ for some nonzero $\phi\in X^*$. There is some $y\in Y$ such that $\phi(y)\neq0$. Define $P:X\to X$ by
$$Px=x-\frac{\phi(x)}{\phi(y)}y.$$
Then $P$ is linear and bounded, $PX=Z$, and $PY\subset Y\cap Z$. If now $z\in Z$, there is a sequence $\{y_n\}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The case for left and right adjoint functors commuting with colimits and limits respectively when the functors are presheaves. Presheaves $F:\mathcal{C}^{opp} \rightarrow \mathcal{D}$, $G:\mathcal{D}^{opp} \rightarrow \mathcal{C}$ are called left (resp. right) adjoint, if there exist a natural bijection (in $C \in \tex... | A functor $G:\mathcal{D}^{op}\to\mathcal{C}$ can instead be considered as a functor $G^{op}:\mathcal{D}\to\mathcal{C}^{op}$, and $F$ and $G$ are left adjoint iff $F$ is left adjoint to $G^{op}$ in the usual sense. So this means $F$ preserves colimits and $G^{op}$ preserves limits, and the latter condition is equivalen... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The longer the base, the longer the hypotenuse Excuse me if this is a silly question, but my plane geometry is very rusty. When I re-read Jack D'Aurizio's answer to the question "How can we prove that $\pi > 3$ using this definition", I wondered why, when viewed from the perspective of high school plane geometry, the a... | Let extend $PX$ and take point $X_1$ such as $PX_1>PX$. Then it's easy to see that $\angle{AXX_1}$ is obtuse and $AX_1$ is the biggest side in $\triangle{AXX_1}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Sub sub sequences and a relation between convergence in probability and a.s convergence I am trying to understand an answer to this question (specifically Siméon's answer) Convergence in probability of the product of two random variables
Im struggling with the statement "$X_{n}$ tends to X in probability if and only i... | I'll list off three main results that are useful to prove this result. If you've seen these before then the result you which to prove is pretty simple. However if you haven't, I've included proofs of the results.
Claim 1: "$X_n$ tends to $X$ in probability if every subsequence of $X_n$ has a sub-subsequence that tends ... | {
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Given $1\leq aA question that was in my calculus 1 test, I remember the question from the mind, I'm pretty sure I didn't miss anything:
Given $1\leq a<b\leq2$, Prove that $$\frac{log(b)-log(a)}{b^2-a^2}<\frac{1}{2}$$
I would like to know the correct form to solve these kinds of questions.
What I did is that I showed... | "Calculus" is the key word. :)
By the mean value theorem: there is $c\in(a,b)$ such that
$$
\frac{\log(b)-\log(a)}{b-a}\frac{1}{b+a}=\frac{1}{c}\frac{1}{b+a}<\frac{1}{1}\frac{1}{1+1}=\frac{1}{2}.
$$
The inequality is because $c>a\geq 1$ and $b+a>a+a\geq 1+1$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Stuck on a recursively defined converging sequence problem. I am given a simple quadratic equation
$$x^2-x-c=0, x>0, c>0$$
and then we define a sequence $\{x_n\}$ with $x_1>0$ fixed and then, if $n$ is an index for which $x_n$ has been defined, we define
$$x_{n+1}=\sqrt{c+x_n}$$.
With that I am asked to prove that $\{x... | Hint
$$x_{n+1}^2-x_n=c$$
$$x_{n+2}^2-x_{n+1}=c$$
so,
$$x_{n+2}^2-x_{n+1}^2-(x_{n+1}-x_n)=0$$
$$(x_{n+2}-x_{n+1})(x_{n+2}+x_{n+1})=(x_{n+1}-x_n)$$
suppose that $x_{N+1}<x_{N}$ for some $N$. What can you conclude?
After that, you have to study the relation between $x_1$ and $x_2$, which will depends on $c$.
| {
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Uniform continuity of $f(x) = x$ on $\Bbb R$ I keep being told how trivial uniform continuity is for $f(x) = x$ on $\Bbb R$, but sometimes those simplest things are hardest for me to see. I haven't been able to find a proof for this anywhere, most likely BECAUSE it is just so simple... in lecture it was given to us as ... | Let $\epsilon>0$. What we wish to show that there exists a $\delta >0$ such that when we assume $|x-y| < \delta$, this implies $|f(x)-f(y)|< \epsilon$. But, note that
$$|f(x) - f(y)|=|x-y|.$$
So the "trivial" part is that we just choose $\delta = \epsilon$, because then we have
$$|x-y| < \delta=\epsilon \implies |x-y|<... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2733121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove or disprove $ABA^{T} = AC$ implies $BA^{T} =C$ Suppose A is $m \times n$ non-zero matrix, $B$ is $n \times n$ of full rank and $C$ is $n \times m$ of full column rank. Can we prove or disprove that $ABA^{T} = AC$ implies that $C = BA^{T}$.
Note: Only $B$ is a square matrix. Rest are rectangular matrices with $m <... | $A=\begin{pmatrix}1 & 0\\
0 & 0
\end{pmatrix}$, $B=C=I_{2}$
$ABA^{T}=\begin{pmatrix}1 & 0\\
0 & 0
\end{pmatrix}=AC$ but $BA^{T}=\begin{pmatrix}1 & 0\\
0 & 0
\end{pmatrix}\neq C$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2733217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Domains of higher powers of two unbounded self-adjoint operators Let $A : D(A) \rightarrow H$ and $B : D(B) \rightarrow H$ be two unbounded self-adjoint operators that are densely defined on a Hilbert space $H$. Suppose we know that $D(A) = D(B)$. Are there sufficient conditions for the equality $D(A^k) = D(B^k)$ to be... | I realized there is an example why this could be generally hard.
Let $A :D(A) \rightarrow L^2(0,1)$ with $D(A) = \lbrace u \in H^2(0,1); u_x(0)=u_x(1)=0 \rbrace$ where $H^2(0,1)$ is the set of elements $L^2(0,1)$ that are least twice weakly differentiable. Additionally, for an element $a \in L^{\infty}(0,1)$ we defin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2733356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Using complex exponential to show the indefinite integration of sin(x)sinh(x) dx Use the complex exponential to evaluate the indefinite integral of
$\sin x \sinh x$.
Express your answer in terms of trigonometric and/or hyperbolic functions
The attached photo is what I have tried so far
| Start with $\sin x=\Im e^{ix}.$ Then your integral is the imaginary part of the following integral:
$$\int e^{ix}\cdot\frac{e^x-e^{-x}}2\, dx=\frac 12 e^{ix}\cdot\left(\frac {e^x}{i+1}-\frac{e^{-x}}{i-1}\right)=\frac 12e^{ix}\cdot \frac{-(e^{-x}+e^x) + i(e^x-e^{-x})}{-2}=\\=\frac 12e^{ix}\cdot (\cosh x-i\sinh x).$$
As ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2733525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Classifying 2-dimensional representations of $S_3$ by hand following Fulton & Harris In §1.3 of Fulton & Harris the authors guide the reader through a classification of the linear representations of $S_3$. Given a representation $\rho:S_3\to \mathrm{GL}(W)$ they first restrict to the abelian subgroup $A_3$ and use the ... | The $\alpha_i$ on page 9 don't have to be distinct, so the eigenspaces don't have to be one-dimensional, a priori (and, in fact, since $W$ is an arbitrary representation there, they will not be one-dimensional in general).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2733776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Joint pdf of uniform dependent random variables Take $n$ non-negative dependent random variables $X_1,...,X_n$ with $Pr(X_i \leq t) = t, t\in[0,1]$ for every $i$ (uniform marginal distributions).
What is an example of a joint pdf for $X_1,...,X_n$ (with the given common marginal distribution), such that $E[\min_i X_i]... | Assume that $(X_1,X_2,\ldots,X_n)$ is a solution and let $M=\min\{X_1,X_2,\ldots,X_n\}$.
Then, for every $i$, $M\leqslant X_i$ almost surely and $\mathbb E(M)=\frac12=\mathbb E(X_i)$ hence $M=X_i$ almost surely. This holds for every $i$ hence $X_1=X_2=\cdots=X_n$ almost surely.
In which case, naturally, $\mathbb E(M)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2733919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Sum and Squares of Uniform Random Variables Suppose I have $X\sim U(-1,1)$ and $Z\sim U(0,0.1)$ independently and want to work out the pdf of a new r.v. defined as such: $Y=X^2+Z$. How could I calculate its distribution? I have tried to work it out but seem to have contradictory things appearing.
I have been able to sh... | I prefer to handle cumulative distribution functions, rather like this:
For $0\le y \le 0.1$ you can say $$\mathbb P(Y \le y) = \int_{z=0}^y 10\, \mathbb P(X^2 \le y-z) \,dz = \int_{z=0}^y 10\, \sqrt{y-z} \,dz = \frac{20}{3}y^{3/2}$$
and taking the derivative will give you $f_Y(y)=10 \sqrt{y}$.
For $0.1\le y \le 1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2734042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Algebraic ideals inside $A \otimes \mathbb K$ Let $A$ be a unital C*-algebra and denote by $F(H)$ the algebraic ideal of finite rank operators on $H = \ell^2(\mathbb N)$. Is is true that
$$
A \odot F(H)
$$
is an algebraic ideal inside $A \otimes \mathbb K$ ? Here $A \odot F(H)$ denotes the algebraic tensor product of... | I think your intuition is correct, even in the case of Abelian $C^\ast$-algebras. Take $A=c_0(\mathcal{Z})$. We have
$$
A \otimes K(H) \simeq c_0(\mathbb{N}; K(H)).
$$
For any element $x=(x_i)_i \in c_0(\mathbb{N}; K(H))$ we can define the following quantity
$$
\mathrm{Im}(x) = \overline{\mathrm{span}}\big\{ \bigcu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2734118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
An affine bundle has a global section? Let $X$ be a manifold. We say $\pi: Y \longrightarrow X$ is an rank $n$ affine bundle if there is an open cover $\{ U_\alpha \}$ of $X$ such that
$ Y \big|_{U_\alpha} \cong U_\alpha \times \mathbb{R}^n $ and the transition function from $U_\alpha$ to $U_\beta$ is given by
$$ (x,v... | Multiplication by functions doesn’t in general make sense, but affine combinations of sections do.
Specifically, if $\sigma,\tau\in\Gamma(\pi)$ are sections and $f, g\in\mathrm C^\infty(X)$ are smooth functions on the base summing to unity, then the obvious definition of $f\sigma + g\tau$ is independent of trivializati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2734237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
minimum number $\alpha$ such that for all $x$: $\alpha^x \geqslant x$ This is a short question which I'm asking just out of curiosity:
Find the minimum positive number $\alpha \in \mathbb{R}$ such that $\forall x \in \mathbb{R}: \alpha^x \geqslant x$.
Maybe somebody even knows how it relates to other mathematical con... | This is only to summarize the answers:
The inequality holds for all $x \leqslant 0$ irrespectively to $\alpha$.
Assuming $x > 0$ we can apply the following transformations:
\begin{gather*}
\alpha^x > x \\
x\ln{\alpha} > \ln{x} \\
\ln{\alpha} > \frac{\ln{x}}{x} \\
\alpha > e^{\frac{\ln{x}}{x}}
\end{gather*}
If we want $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2734518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Proving that if $\lim\limits_{n\to\infty}a_n=+\infty$ and $\{b_n\}$ is a bounded sequence, then $\lim\limits_{n\to\infty}(a_n+b_n)=+\infty$ I am trying to prove the following problem
If $\lim\limits_{n\to\infty}a_n=+\infty$ and $\{b_n\}$ is a bounded sequence, then $\lim\limits_{n\to\infty}(a_n+b_n)=+\infty$
I have ... | Let $B$ be the bound for $(b_n)$. For any $M\gt0$ choose $n_0\in\mathbb N$ such that $n\ge n_0 \implies a_n\gt M+B$. Then $a_n+b_n\gt M$...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2734678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
chain equivalence between cubical chain complex and simplicial chain complex I remember hearing before that for a topological space, the cubical chain complex and the simplicial chain complex are chain equivalent. Is this true? If yes, can someone provide me with a reference where I could see how this chain equivalence... | This equivalence was first proved in the classical paper
Eilenberg, S., and Mac Lane, S., Acyclic models. Amer. J. Math. 75 (1953)
189–199.
The method of acyclic models was extended from chain complexes to crossed complexes in Section 10.4 of the book Nonabelian Algebraic Topology. The advantage of the more genera... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2734828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show there are infinitely many positive integers $k$ such that $\phi(n)=k$ has exactly two solutions where $n$ is a positive integer
Show that there are infinitely many positive integers $k$ such that the equation $\phi(n)=k$ has exactly two solutions, where $n$ is a positive integer.
Not entirely sure where to start... | Let $k = 2 \times 3^{6m + 1}$. We claim that if $m > 0$, then $\phi(n) = k$ has exactly $2$ solutions. (For $m = 0$, it has four solutions: $7$, $9$, $14$, and $18$.)
Let $n = p_1^{a_1} p_2^{a_2} \cdots p_i^{a_i}$, where $p_j$ is a prime number, and $a_j$ is a positive natural number. Then $n$ is a solution if and only... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2734926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
For a group $G$ such that $|G| = p^3$, for $p$ prime, either $|Z(G)| = p$ or $G$ is abelian. I came with this proof and I found some other proofs online, but mine is different and I want to see if I made any mistakes.
Problem:
Suppose $|G| = p^3$, where $p$ is a prime. Show that either $|Z(G)|=p$ or $G$ is abelian.
Cas... | Suppose that $G$ is non-abelian.
We know that $Z(G)\leq G$.
By Lagrange's Theorem $|Z(G)|$ must divide $|G|$.
Since $|G|=p^{3}$ the only possibilities are $1, p, p^{2}, p^{3}$.
$|Z(G)|\neq p^{3}$ because otherwise we will have $Z(G)=G$ but $G$ is non-abelian.
$|Z(G)|\neq p^{2}$ also because otherwise we will have the o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2735078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Minimum value trace of A*Atranspose For any matrix $A$, let $A^T$ denote its transpose matrix.
What is the minimum value of $\mathrm{trace}(AA^T)$ for an $n \times n$ non-singular matrix $A$ with integer entries?
| When $A$ is nonsingular, $AA^T$ is positive definite. Hence it has a positive diagonal. But $AA^T$ is also an integer matrix. Hence its diagonal entries are positive integers, meaning that $\operatorname{tr}(AA^T)\ge n$. Obviously, tie occurs when $A=I$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2735205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Is there an example of a, non-Hausdorff, topological vector space which has bounded subspaces Let $E$ be a topological vector space which is not Hausdorff.
Is it true that every non-trivial subspace $M$ of $E$ is necessarily unbounded?
Or, does there exist, non-Hausdorff, topological vector spaces containing non-triv... | Take $\mathbb{R}^2$ with the semi-norm $$\|(x,y) \| := |x|$$ and the subspace $$Y:= \{(0,y) : y \in \mathbb{R} \}, $$ then $( \mathbb{R}^2, \| \cdot \|)$ is a non-Hausdorff TVS with bounded subspace $Y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2735305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why can't I cancel $2x-3$ from $(2x-3)(x+5)=9(2x-3)$? Why are these simplifications wrong?
$$\begin{align}
(2x-3)(x+5)=9(2x-3) &\quad\to\quad \frac{(2x-3)(x+5)}{2x-3} = \frac{9(2x-3)}{2x-3} \quad\to\quad x+5 = 9\\[4pt]
x(x+2)=x(-x+3) &\quad\to\quad \frac{x(x+2)}{x} = \frac{x(-x+3)}{x} \quad\to\quad x+2=-x+3
\end{align}... | From here
$$(2x-3)(x+5)=9(2x-3)$$
we can observe that $2x-3=0$ is a solution and for $2x-3\neq 0$ we can cancel out and obtain
$$(2x-3)(x+5)=9(2x-3)\iff x+5=9\iff x=4$$
thus the solutions for the original equation are $x=\frac32$ and $x=4$.
As an alternative note that
$$(2x-3)(x+5)=9(2x-3)\iff 2x^2+7x-15=18x-27 \iff2x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2735413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
The Chain Rule - finding differentials I have been given:
*
*$V=f(x,y,z)$, with $x= r\cos\theta$, $y= r\sin\theta$ , and $z=t$.
*And asked, find $dV/dr$, $dV/d\theta$ and $dV/dt$
Would
$$
\frac{dV}{dr} = \frac{dV}{dx} * \frac{dx}{dr} + \frac{dV}{dy} * \frac{dy}{dr}
$$
?
If so, what is $dV/dx$ if I have just bee... | First: using the same name for different things is a bad habit that causes confusion. Use different names for different things. In your case:
$$V(r,\theta,t) = f(r\cos\theta,r\sin\theta,t).$$
Then:
$$
\frac{\partial V}{\partial r} =
\frac{\partial f}{\partial x}\frac{\partial x}{\partial r} +
\frac{\partial f}{\partial... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2735523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Properties of a cumulative distribution function and rescaling Consider a cumulative distribution function $F$.
Take $a: (0,\infty) \rightarrow (0,\infty)$ and $b: (0,\infty) \rightarrow (-\infty,\infty)$. Let $a_s\equiv a(s)$ and $b_s\equiv b(s)$ for all $s \in (0,\infty)$.
Assume
$$
[F(a_sx+b_s)]^s=F(x), \quad\foral... | Because $$
(F(a_s x + b_s))^s = F(x), \quad \forall s > 0, x \in \mathbb{R}
$$
then for any $u > 0$, $y \in \mathbb{R}$, take $(s, x) = \left( u, \dfrac{y - b_u}{a_u} \right)$ to get$$
(F(y))^u = F\left( \frac{y - b_u}{a_u} \right).
$$
Now, for any $s, t > 0$, $x \in \mathbb{R}$, take $(u, y) = (s, a_{st} x + b_{st})$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2735690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Axis Rotation to determine new points I am looking at this answer .
https://math.stackexchange.com/a/62248/474907, but the formula
c = cos(a); // compute trig. functions only once
s = sin(a);
xr = xt * c - yt * s;
yr = xt * s + yt * c;
differs from the wikipedia entry. So, I am a little confused, which one is correc... | If you're referring to the sign difference, this seems to be the key sentence in the SE answer:
In 2D graphic libraries the x-axis goes to the right and the y-axis goes down
The wikipedia entry is based on the y-axis going up. The formula is slightly different depending on whether you consider the angle to be clockwi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2735790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
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